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A357259
a(n) is the number of 2 X 2 Euclid-reduced matrices having determinant n.
2
1, 2, 3, 5, 5, 8, 7, 11, 10, 14, 11, 19, 13, 20, 18, 24, 17, 30, 19, 31, 26, 32, 23, 44, 26, 38, 34, 45, 29, 54, 31, 52, 42, 50, 38, 70, 37, 56, 50, 70, 41, 76, 43, 73, 63, 68, 47, 97, 50, 80, 66, 87, 53, 100, 62, 96, 74, 86, 59, 132, 61, 92, 85, 109, 74, 124, 67, 115, 90, 118
OFFSET
1,2
COMMENTS
See Bacher link for the definition of Euclid-reduced.
LINKS
Roland Bacher, Euclid meets Popeye: The Euclidean Algorithm for 2X2 matrices, arXiv:2209.09529 [math.NT], 2022.
Roland Bacher, Euclid meets Popeye: The Euclidean Algorithm for 2 X 2 Matrices, Comptes rendus de l’Académie des sciences, Volume 361 (2023), p. 889-895.
FORMULA
a(n) = Sum_{d|n, d^2>=n} d+1-n/d.
From Ridouane Oudra, Oct 30 2023: (Start)
a(n) = Sum_{d|n} max(d-n/d, 1).
a(n) = ceiling(tau(n)/2) + (1/2)*Sum_{d|n} abs(d-n/d).
a(n) = A038548(n) + A079667(n). (End)
G.f.: Sum_{k>=1} x^(k^2) / (1 - x^k)^2. - Ilya Gutkovskiy, May 17 2024
MAPLE
with(numtheory): seq(add(max(d-n/d, 1), d in divisors(n)), n=1..80); # Ridouane Oudra, Oct 30 2023
MATHEMATICA
a[n_] := DivisorSum[n, # + 1 - n/# &, #^2 >= n &]; Array[a, 100] (* Amiram Eldar, Sep 21 2022 *)
PROG
(PARI) a(n) = sumdiv(n, d, if (d^2 >= n, d+1-n/d));
CROSSREFS
Cf. A357260.
Sequence in context: A267453 A063914 A209187 * A166250 A174088 A304493
KEYWORD
nonn
AUTHOR
Michel Marcus, Sep 21 2022
STATUS
approved